Riemannian 10j symbols. I would appreciate any confirmation of these calculations using other algorithms. If you would like me to calculate some other 10j symbols, just send me an e-mail: jdc at uwo.ca The calculations below were done using the algorithm by Christensen and Egan (although not using the same code that is available from http://jdc.math.uwo.ca/spin-foams). In fact, most of the timings below use an old implementation that is quite poor. See the last group for more up-to-date timing information. Note that some Riemannian 10j symbols have *negative* values. See the last group. ----- The first group consists of 10j symbols all of whose spins are equal. The first column shows twice the spin, and the second column is the value of the regular 10j symbol. For higher spins, these are calculated using 55 decimal digits of precision, but for very high spins round-off error may have reduced the accuracy of the answer. See http://www.gregegan.net/SCIENCE/tenJ/tenJ.html for some exact rational calculations. 0 1.0 1 0.388888888888888888888888888888888888889 2 0.204666666666666666666666666666666666665 3 0.126027210884353741496598639455782312921 4 0.0852897095346074937911672605550156570553 5 0.0614798243054272738873852046393790364095 6 0.0463744479722045280030418435137259436498 7 0.036205685514232112489000456178133645522 8 0.0290443887063886246458211427912998485716 9 0.023817533232487066150352233756036475289 10 0.019889023166009833374130273770664786767 11 0.0168624749284428690332524531452973628357 12 0.01448088549814993868346315515537968071446 13 0.01257218593732248369913899011444167458453 14 0.01101804659078404393829883445196685796377 15 0.009735169351913561892935619617995919703 16 0.00866364058803167555706773157810683891263 17 0.00775946522467552401793579382095093504326 18 0.00698967858518895499762171467681428743088 19 0.00632909971048648675317438025437245648942 20 0.00575815329276568001482230155782714975919 21 0.00526139460422358420983662546632696824606 22 0.00482649665823258654706321484124158375956 23 0.00444353882699164072240372337539540582306 24 0.00410449033431277216791331153725082110067 25 0.0038028200351133860319654441657620073605 26 0.00353319052401389078429667039348940211956 27 0.00329121254199596408115986904612014166607 28 0.00307324670250825315698327214914979324703 29 0.00287624536547637640120401303651443229354 30 0.00269762966250969504971365782879466770073 31 0.00253519675978629916583264564260397790998 32 0.00238705175212598721233101621150244628891 33 0.00225155805976762636014747029400002813986 34 0.00212730035607114621141306762866265806619 35 0.00201305498111669948552069057087102179364 36 0.001907764272704512548613994298834250593834 37 0.0018105128920073665522073688403166032272 38 0.00172050564970820164131327350754339433681 39 0.00163704727705038851946228518460337028344 40 0.001559524936622451502146115168333737310227 41 0.00148739410465344983367583442315119798557 42 0.00142016797647561020895948090708165237536 43 0.00135740999041886637018869638788423384128 44 0.00129872864172799247325826961443726054852 45 0.00124377359071383987484022764866281527544 46 0.00119223217700295170638790551551903058233 47 0.00114382576285686709733276529790982267502 48 0.00109830571967696291937075307707109549286 49 0.00105544921483409652243728727511099860558 50 0.0010150551583145716201159145333530144976 51 9.769406961063534107144387412228251413e-4 52 9.40938514917345990433705448700225812503e-4 53 9.06895017936067236109742852906867240197e-4 54 8.74669225608424314038673114064870588774e-4 55 8.44132117223620929659142920022949326317e-4 56 8.15166094252697232894132551810552418123e-4 57 7.87664311791665985077557771933290410315e-4 58 7.6152975511813344659052440181669536057e-4 59 7.36674083284897185304397786460718726805e-4 60 7.13016372558504947597431851334439379803e-4 61 6.90481938915388709882871829361388011903e-4 62 6.6900139541555614150376908679208666728e-4 63 6.48510025121165182096275010536453783512e-4 64 6.2894745566308408193446810774177786431e-4 65 6.1025754180622243331592569556903037466e-4 66 5.92388322137388508378770346825412967424e-4 67 5.75291923696490040394744318663897007896e-4 68 5.5892433545986442344918057123123380764e-4 69 5.43245037579205836499441633414003330663e-4 70 5.28216534002788837458829820642193751904e-4 71 5.13803872122418417186751571159476594634e-4 72 4.99974235548164214618868959351089152392e-4 73 4.86696668391815474487752773883037090043e-4 74 4.73941944561377421045717897932930213055e-4 75 4.61682550730456034902755658025764794938e-4 76 4.49892722008271381911635038399972482247e-4 77 4.3854846320136240851612819859142108042e-4 78 4.27627505382270365015343594207699177648e-4 79 4.17109179052194894033885052439315238301e-4 80 4.06974219232906869046411700090132195783e-4 81 3.97204542585094212143668363923904385054e-4 82 3.87783044771604784637391881400136558336e-4 83 3.78693456822747527678177373925084743712e-4 84 3.6992027732445902322840039879900379891e-4 85 3.61448771636604199707420801118279011025e-4 86 3.53265009188907586296682324060979856537e-4 87 3.45355901534991279038898323119212413653e-4 88 3.37709208952713658199916974074258015545e-4 89 3.3031349888331922865288197433312433225875935907810851830345e-4 90 3.231580590832291395256368704255035226242033764555460878314e-4 91 3.1623278492775523295564853261124349203949270497999842596276e-4 92 3.095280684800970761015369736177786737688312862260050412555e-4 93 3.030347147310397110959997222843709575005387087173327234722e-4 94 2.967438995764355467743041132745861578016054517024581182047e-4 95 2.9064716917413601238895164227189195076947126904691034032576e-4 96 2.847364668309830225708542337234359546580120113224232546103e-4 97 2.790041660334795216571609132627929821857503875512585031398e-4 98 2.734430887140340462589357754813794785284355402151126164918e-4 99 2.680464954565406991097940541627916025922102201337822622152e-4 100 2.628080458320912201798192877084863414221181069804311630089e-4 101 2.577217381126484725262045121452694730377695052672210835903e-4 102 2.527818444575775323391018888047426778053877923052243082914e-4 103 2.479828583051540312189636044066912157592242723220463020997e-4 104 2.433194654440667190632918597968247254460247819166208757492e-4 105 2.387865414131365927160891159378493506577753421410757154813e-4 106 2.3437916884324376083039514827504443655532693633688511310478e-4 107 2.3009266226641022618512242679397494003930177543179061373407e-4 108 2.259225866447431931724774479472042483506615319085704996058e-4 109 2.2186475946563265870964623415349193964121665294736937086696e-4 110 2.1791523306745150269147332492201090013870486312636268830788e-4 111 2.140702612741625051236712128015541140467106884803883086147e-4 112 2.1032625980276949235449530117404112431334934097481585369294e-4 113 2.066797715518820545408095376371498297189623169588592316866e-4 114 2.0312744550463341469347444504194639546182928402926940020003e-4 115 1.9966603272568018558496952693879429108017195157808231754607e-4 116 1.9629239685411087825001457575534847755687131625260483650865e-4 117 1.9300353177551308189362521997987211749713007334601217066263e-4 118 1.8979657734166291948796819613309472150522785600884149425806e-4 119 1.8666882551159964621853561725363885629543335137332647197672e-4 120 1.836177133408837953413557172741447188997935702871994529562e-4 121 1.806408042427018877408654678983163290209601968477363642306e-4 122 1.7773576306962135205195539021750858176159892936568882232267e-4 123 1.749003324392469057457630060157386395124432747346540213935e-4 124 1.7213231685765283791143422121245473147318003521552957042944e-4 125 1.6942957808239009345954962337789810469627457776009563011694e-4 126 1.6679004106170143480390836551304108067420755477612834480077e-4 127 1.6421170619709833518677050663745981313041732207985967201053e-4 128 1.6169266182527216669043730587727000810361110346568395747193e-4 129 1.5923109123143929618076359630156058539824566018477397748022e-4 130 1.5682527088798653567125433392948710753939684372497300412905e-4 131 1.544735599898483422100489428933653730345758560270416816385e-4 133 1.4992622052972621213608087323478452521969139293212717216258e-4 134 1.4772758239632607762642698763787291387870512720254222688963e-4 135 1.4557701768109641859572098708291913102082435901808663694643e-4 137 1.4141448909188402415353632435846431840520881903402173830194e-4 138 1.3939983690984068975180073741385853633626910412096520105254e-4 139 1.3742789776482534015585725891983536424231078215117537610474e-4 140 1.3549747735208288882698221043180416472617917928594290182754e-4 141 1.336074374497198334688539894487471089042767394252908487584e-4 142 1.3175668558389496806646238277082867716613458462434781995446e-4 143 1.2994416334280342708268853870664949581844344714013121261858e-4 145 1.2642969287851153839876446262663647476860771651930368763518e-4 146 1.247257388130748113182976855993842257090476206984378084605e-4 147 1.2305600944716189323256707969190204971585842244319457400604e-4 148 1.2141957363523839336272173688680048304822974830365278278144e-4 149 1.1981554021860261126636184483910053362426560387660061865547e-4 150 1.1824305996280494834680465303895582849240572417611633082656e-4 151 1.1670132274881826895252456969419386294844355531125437580053e-4 152 1.1518955089043841757605518930804755894552527374968447561325e-4 153 1.1370699079018494129576194069965882481945202197057121644352e-4 154 1.1225290562398811981081634334338378788126623838756518470935e-4 155 1.1082657124197459980741821417881709253001658630143987345497e-4 156 1.0942727622301614851072719621188731129833969635278834945438e-4 157 1.0805432550924167552655683899634295166812742531517558101089e-4 158 1.0670704582331398095005619676740340877075506156519468571704e-4 159 1.053847905528915131071672693888160081273937432927368541371e-4 160 1.0408694211747314393481363667583476918718439297310252074637e-4 161 1.028129108483782044356433203914880190856015266194736962853e-4 162 1.0156213071529299676638709329655584356767048175754409198228e-4 163 1.0033405334521408847050835015273859772409284503297833630817e-4 164 9.91281423178499782466872454113630303137370363689176689891e-5 165 9.79438695259825813960378269439267464831266687675877765482e-5 166 9.67807145689263696367392282525497534917858853594007335897e-5 167 9.563816702659990165686026180740031515453463578650221077693e-5 168 9.451573045446864643412093340062799876865711402032958224482e-5 169 9.341292639718382493900085245497658143462997551564847634358e-5 170 9.232929680918141981269657049920038752753208090561450910253e-5 171 9.126440392856237480316800282285726836781121238943065724024e-5 172 9.021782761301967844082202137888383446879176026027329895227e-5 173 8.918916105582994097888181485118284838878199591933583896252e-5 174 8.81780063357404634682482081249104152091026310533312640841e-5 175 8.718397124571954185922031927950325655705922494237751169883e-5 176 8.62066683229018848076391370546801680592660021401155889755e-5 177 8.524571617745653496701498209515976025827549874992391568038e-5 178 8.43007423999149160208643621213304053246336713868847949942e-5 179 8.337138680535108596987236770612528719976183082678888789147e-5 180 8.245730371927117073186776016938154312742636296664672096487e-5 181 8.155816242003893433624060347969779847952253578361062551745e-5 182 8.067364555745714110633793851273429632029212139145585378866e-5 183 7.98034461015053505480453778601444491693596089242918717529e-5 184 7.894726387639491883394736162759011050485405353321647691541e-5 185 7.81048028359510282500975690287928166604053249199919516393e-5 186 7.727576992026822160164931933598978270551288304986024354444e-5 187 7.645987573315857582950776428203093894482144247207268829186e-5 188 7.565683662253965064417092413550513949770704890719607054282e-5 189 7.486637726847192065733570461881108529313054859326492066227e-5 190 7.408823274766966307556975450761560910656260103954451430879e-5 191 7.332214928136809083148349610019248707469884903104512771197e-5 192 7.256788338313041257849302141210755334859947701765903371783e-5 193 7.182519971022017810483609783436313178860538105465427227435e-5 194 7.109386837368975284184851046026579081997109548534461016673e-5 195 7.037366262358306976027374203348223485795893449461712805194e-5 196 6.96643576524022615141054209597274129752419660968166893864e-5 197 6.896573082926984460032385991681660221558288145416119747288e-5 198 6.827756315434850059341959482150005061222136786381840303078e-5 199 6.759964129876903657467923733411365175206635812824220614385e-5 200 6.693175941655875570233399944066600068277726805661385103657e-5 201 6.62737200347010939248483573549303606288326749334352632694e-5 202 6.562533368954288515733800936383883412647756341496329231292e-5 205 6.373628378387612139176332601611650123450497620833160375673e-5 206 6.312471572121172650023651868720649509319308220514365764438e-5 207 6.252191521665056272943754225357578220675796074820323757182e-5 210 6.076442826774379814606289500899230503730398011947736614233e-5 212 5.963359796815776759746908294235261896738909292157862259156e-5 215 5.799564601812751997708518009181547742168439160961568209996e-5 217 5.694092002977509522443540556274032528969459084190076513013e-5 220 5.541198628587535057534261446540005081438006275734417477753e-5 225 5.299717670900908337507863422424400239030307767977670870619e-5 230 5.073693157097355205587698360954811148913961828603577403884e-5 235 4.861823064441222874172884683014803170150341154138322604516e-5 240 4.662958910234209942877947050153995016655503934908469797953e-5 245 4.476049078011693170513574804498238647450223417575353518558e-5 250 4.300161623670619024561958937233969502641193408554198716682e-5 255 4.134439855540520354319407317817183765554193443733017148738e-5 260 3.978121026628326079863688487441670493300898584398777406728e-5 265 3.8305017093243171685306871982006839866114233974266028792625e-5 300 2.9940162142032601925591883196498779884538368659804953899604e-5 End of regular 10j symbols. ----- The second group of 10j symbols consists of multiples of the following ten input spins: 1 3/2 1/2 1 1/2 1/2 3/2 1/2 1 1 Answer: 0.0416666666666666643537. Multiple: 1. Time: 0 ms. Answer: 0.0088772270813087146568. Multiple: 2. Time: 0 ms. Answer: 0.0032042975920526942449. Multiple: 3. Time: 10 ms. Answer: 0.0014990339735524618302. Multiple: 4. Time: 20 ms. Answer: 0.0008180161177295162088. Multiple: 5. Time: 50 ms. Answer: 0.0004943102250590122003. Multiple: 6. Time: 100 ms. Answer: 0.0003211719998838924038. Multiple: 7. Time: 200 ms. Answer: 0.0002203103293439064318. Multiple: 8. Time: 350 ms. Answer: 0.0001576231702278443176. Multiple: 9. Time: 610 ms. Answer: 0.0001166312244500585573. Multiple: 10. Time: 1000 ms. Answer: 0.0000887053783363668183. Multiple: 11. Time: 1620 ms. Answer: 0.0000690282513956160117. Multiple: 12. Time: 2460 ms. Answer: 0.0000547659079913130254. Multiple: 13. Time: 3620 ms. Answer: 0.0000441768951717394742. Multiple: 14. Time: 5200 ms. Answer: 0.0000361508396117670859. Multiple: 15. Time: 7370 ms. Answer: 0.0000299569698373605491. Multiple: 16. Time: 10120 ms. Answer: 0.0000251009447049803848. Multiple: 17. Time: 13890 ms. Answer: 0.0000212402142804200475. Multiple: 18. Time: 18400 ms. Answer: 0.0000181323111646042982. Multiple: 19. Time: 24440 ms. Answer: 0.0000156023319380775333. Multiple: 20. Time: 31810 ms. Answer: 0.0000135219543836963565. Multiple: 21. Time: 41270 ms. Answer: 0.0000117955835179612312. Multiple: 22. Time: 52080 ms. Answer: 0.0000103510134344362871. Multiple: 23. Time: 66100 ms. Answer: 0.0000091330142420718405. Multiple: 24. Time: 82100 ms. Answer: 0.0000080988523852257496. Multiple: 25. Time: 102250 ms. Answer: 0.0000072151125747708286. Multiple: 26. Time: 125910 ms. Answer: 0.0000064554108586767923. Multiple: 27. Time: 154330 ms. Answer: 0.0000057987272919911887. Multiple: 28. Time: 187290 ms. Answer: 0.0000052281755739440822. Multiple: 29. Time: 228020 ms. Answer: 0.0000047300849269413945. Multiple: 30. Time: 270560 ms. 270 seconds is 4 1/2 minutes. ----- The third group of 10j symbols consists of multiples of the following ten input spins: 1 1 1 1 1 1/2 1/2 1/2 1/2 1/2 This is run with a better implementation and a faster cpu. Note how much smaller the times are. And note that some answers are negative. Answer: -0.0943287037037036923959. Multiple: 1. Time: 0.037 ms. Answer: 0.0311311953352769663073. Multiple: 2. Time: 0.137 ms. Answer: -0.0152448524052478124463. Multiple: 3. Time: 0.372 ms. Answer: 0.0090136022510749116282. Multiple: 4. Time: 0.822 ms. Answer: -0.0059427743834604643963. Multiple: 5. Time: 1.63 ms. Answer: 0.0042059019670120526349. Multiple: 6. Time: 2.81 ms. Answer: -0.0031302721230514899201. Multiple: 7. Time: 4.71 ms. Answer: 0.0024198825173873364687. Multiple: 8. Time: 7.31 ms. Answer: -0.0019268999197411462374. Multiple: 9. Time: 11.3 ms. Answer: 0.0015707067775126318070. Multiple: 10. Time: 15.8 ms. Answer: 0.0004034541530169647437. Multiple: 20. Time: 215 ms. Answer: 0.0001809110529926336269. Multiple: 30. Time: 1080 ms. Answer: 0.0001022115435021501604. Multiple: 40. Time: 3610 ms. Answer: 0.0000655876894503488024. Multiple: 50. Time: 9250 ms. Answer: 0.0000456267106893390502. Multiple: 60. Time: 20500 ms. Answer: 0.0000257209965927777247. Multiple: 80. Time: 73970 ms. Answer: -0.0000250918181271622359. Multiple: 81. Time: 78160 ms.